There is an obligation to recite Hallel (Tehillim chapters 113-118) on Sukkot, Shemini Atzeret, Chanukah, the first day(s) of Pesach, and Shavuot.
We also recite the Hallel on Chol HaMoed Sukkot, because each day has a different korban, and thus necessitates Hallel. Likewise, Hallel is recited on days when the Lulav is taken, for the Torah refers to each of those days as chag (holiday). On Chol HaMoed Pesach, we only recite half-Hallel, since the same korban is brought every day.
Days when we recite Hallel are called Mo'ed, and it is forbidden to do melacha. Exceptions to this are the days of Chanukah, when even though they are not sanctified as a chag, however we recite Hallel due to the miracle(s) that occurred.
On Rosh Chodesh, however, there is no prohibition of melacha, and therefore there is not a mitzvah to recite Hallel. However, there is an ancient custom to recite Hallel on Rosh Chodesh as a remembrance for the mitzvah of Kiddush HaChodesh, &/or to publicize that it is Rosh Chodesh. In order to emphasize that this is done as a custom, we skip parts of the Hallel.
בְּעָא מִינֵּיהּ אַחַי תַּנָּא דְבֵי רַבִּי חִיָּיא מֵרַבִּי חִיָּיא: בְּהַלֵּל וּבַמְגִילָּה מַהוּ שֶׁיַּפְסִיק? אָמְרִינַן קַל וָחוֹמֶר: קְרִיאַת שְׁמַע דְּאוֹרָיְיתָא — פּוֹסֵק, הַלֵּל דְּרַבָּנַן מִבַּעְיָא?! אוֹ דִלְמָא פַּרְסוֹמֵי נִיסָּא עֲדִיף.
אֲמַר לֵיהּ: פּוֹסֵק, וְאֵין בְּכָךְ כְּלוּם. אָמַר רַבָּה: יָמִים שֶׁהַיָּחִיד גּוֹמֵר בָּהֶן אֶת הַהַלֵּל, בֵּין פֶּרֶק לְפֶרֶק פּוֹסֵק, בְּאֶמְצַע הַפֶּרֶק — אֵינוֹ פּוֹסֵק, וְיָמִים שֶׁאֵין הַיָּחִיד גּוֹמֵר בָּהֶן אֶת הַהַלֵּל — אֲפִילּוּ בְּאֶמְצַע הַפֶּרֶק פּוֹסֵק.
Aḥai, the tanna who recited mishnayot in the school of Rabbi Ḥiyya, raised a dilemma before Rabbi Ḥiyya: May one interrupt during the recitation of hallel and the reading of the Megilla, Esther, to greet someone? Do we say that it is an a fortiori inference; if in the middle of Shema, which is a biblical obligation, one may interrupt in order to greet someone, all the more so, in the middle of hallel, which is a rabbinic obligation, one may interrupt? Or, perhaps, publicizing the miracle is more significant, so one may not interrupt hallel or the Megilla at all.
Rabbi Ḥiyya said to him: One interrupts and it is of no concern. Rabba said: On days when the individual completes the entire hallel, i.e., the days on which there is a rabbinic obligation to recite hallel, an individual may interrupt between one paragraph and another; however, one may not interrupt in the middle of the paragraph. On days where the individual does not complete the entire hallel, i.e., days on which the recitation of hallel is merely a custom, not a rabbinic obligation, one may interrupt even in the middle of the paragraph.
ובראש חודש אפילו צבור אין קורין אותו אלא משום מנהגא זכר לקידוש החודש.
On Rosh Chodesh, even the communal reading of Hallel is only a minhag, as a remembrance for the mitzvah of Kiddush HaChodesh (sanctification of the new month).
...דימים דאין גומרין בהן הלל אין חובה לאמרו ומה שאנו אומרין אותו היינו משום מנהג. וכתוב במחזור ויטרי כיון שאינו אלא מנהג בעלמא לא מברכינן עליה כדאמר פרק לולב וערבה (סוכה ד' מד:) אין מברכין על דבר שהוא מנהג. ומיהו אומר ר"ת דאינו ראיה דאטלטול ודאי לא מברכינן אבל על מצוה פשיטא דמברכינן דהא חזינן כל יו"ט שני אינו אלא מנהגא ומברכין ...מברכים בטוב על מנהגא.
...On the days when we do not recite the entire Hallel, the reason is because on those days there is no obligation, rather, it is a minhag (custom). It is written in Machzor Vitri that since it's only a minhag, we do not recite a blessing over it, as stated in the Talmud (Sukkah 44b): One does not recite a blessing over something that is a minhag [i.e., the taking of the Aravah (willow) branch on Hoshana Rabba].
However, Rabbeinu Tam says that the fact that we don't recite a blessing over taking the Aravah is not a proof [that we should not recite a blessing over Hallel on Rosh Chodesh], because we do not recite a blessing merely for picking up a willow branch! However, one should certainly recite a blessing over a mitzvah, even if it merely a custom and not an obligation. For in fact we see that we recite a blessing sanctifying the 2nd day of Yom Tov, even though the 2nd day in the diaspora is also a minhag, and it certainly is appropriate to recite a blessing over a minhag.
[In contrast,] the recitation of Hallel on Rosh Chodesh is a custom and not a mitzvah. It is observed [only] communally. [To emphasize that it is a custom,] passages are skipped when it is read. A blessing should not be recited over [this reading], since a blessing is not recited over a custom.
A person [praying] alone should not recite [the Hallel] at all [on Rosh Chodesh]. If, however, he began its recitation, he should complete it, skipping the passages the community would skip as he reads it.
Similarly, on the other days of Pesach, [the Hallel] is read while skipping passages.
...ודעת רש"י דצבור נמי אין מברכין עליו וכ"ד הרמב"ם בהלכות חנוכה
The opinion or Rashi and Rambam is that one does not recite a blessing [over Hallel on Rosh Chodesh].